Finding The Vertex Of The Quadratic Function F(x) = X² - 8x - 9

by ADMIN 64 views

In this article, we will delve into the process of determining the vertex of the quadratic function f(x) = x² - 8x - 9. Understanding the vertex is crucial for grasping the behavior and characteristics of parabolas, the graphical representation of quadratic functions. The vertex represents the point where the parabola changes direction, either reaching its minimum value (if the parabola opens upwards) or its maximum value (if the parabola opens downwards). We will explore the formula for calculating the vertex and apply it to the given function, providing a step-by-step explanation to ensure clarity.

Understanding Quadratic Functions and the Vertex

A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The vertex of the parabola is a significant point as it represents the extremum (minimum or maximum) of the function. The coefficient a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0).

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. This form provides a direct way to identify the vertex. However, quadratic functions are often presented in the standard form f(x) = ax² + bx + c. In this case, we can use a formula to calculate the vertex coordinates. The x-coordinate of the vertex is given by -b / 2a, and the y-coordinate is found by substituting this x-value back into the original function, i.e., f(-b / 2a).

Calculating the Vertex of f(x) = x² - 8x - 9

Now, let's apply this knowledge to the given quadratic function f(x) = x² - 8x - 9. First, we need to identify the coefficients a, b, and c. Comparing the function to the standard form f(x) = ax² + bx + c, we can see that:

  • a = 1
  • b = -8
  • c = -9

Using the formula for the x-coordinate of the vertex, we have:

x = -b / 2a = -(-8) / (2 * 1) = 8 / 2 = 4

Next, we substitute this x-value (4) back into the original function to find the y-coordinate of the vertex:

f(4) = (4)² - 8(4) - 9 = 16 - 32 - 9 = -25

Therefore, the vertex of the quadratic function f(x) = x² - 8x - 9 is (4, -25). This means that the parabola reaches its minimum value at the point (4, -25), as the coefficient a is positive (1), indicating that the parabola opens upwards.

Significance of the Vertex

The vertex is a crucial point for understanding the behavior of a quadratic function and its corresponding parabola. Here's why:

  1. Extremum: The vertex represents the minimum or maximum value of the function. If the parabola opens upwards (a > 0), the vertex is the minimum point. If the parabola opens downwards (a < 0), the vertex is the maximum point. In our example, the vertex (4, -25) is the minimum point of the function f(x) = x² - 8x - 9.
  2. Axis of Symmetry: The vertical line passing through the vertex is called the axis of symmetry. It divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. For our function, the axis of symmetry is x = 4.
  3. Range: The y-coordinate of the vertex determines the range of the quadratic function. If the parabola opens upwards, the range is [k, ∞), where k is the y-coordinate of the vertex. If the parabola opens downwards, the range is (-∞, k]. For our function, the range is [-25, ∞).
  4. Transformations: The vertex form of a quadratic function, f(x) = a(x - h)² + k, reveals the transformations applied to the basic parabola y = x². The value of h represents a horizontal shift, and the value of k represents a vertical shift. In this form, the vertex (h, k) is readily apparent. Understanding these transformations allows us to quickly sketch the graph of the quadratic function.

Visualizing the Parabola

To further solidify our understanding, let's visualize the parabola of the function f(x) = x² - 8x - 9. We know the vertex is at (4, -25), and since the coefficient a is positive, the parabola opens upwards. The axis of symmetry is the vertical line x = 4. We can also find the x-intercepts by setting f(x) = 0 and solving for x:

x² - 8x - 9 = 0

This equation can be factored as:

(x - 9)(x + 1) = 0

So, the x-intercepts are x = 9 and x = -1. The y-intercept is found by setting x = 0:

f(0) = (0)² - 8(0) - 9 = -9

Thus, the y-intercept is (0, -9). With this information, we can sketch a parabola that opens upwards, has its vertex at (4, -25), crosses the x-axis at x = -1 and x = 9, and crosses the y-axis at y = -9. This visual representation helps to confirm our calculations and provides a complete picture of the function's behavior.

Conclusion

In summary, we have successfully determined the vertex of the quadratic function f(x) = x² - 8x - 9 to be (4, -25). We achieved this by applying the formula -b / 2a to find the x-coordinate and then substituting this value back into the function to find the y-coordinate. Understanding the vertex is essential for analyzing quadratic functions and their parabolic graphs, as it provides information about the function's extremum, axis of symmetry, range, and transformations. By mastering this concept, we gain valuable insights into the behavior and characteristics of quadratic functions.

This process of finding the vertex can be applied to any quadratic function in the form f(x) = ax² + bx + c. By accurately identifying the coefficients a, b, and c and applying the formulas, we can confidently determine the vertex and gain a deeper understanding of the quadratic function's properties. Remember that the vertex is a key feature of the parabola and provides valuable information about its shape, position, and behavior. Continue practicing with different quadratic functions to solidify your understanding and enhance your problem-solving skills in this area of mathematics.